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How has Solve changed in 3.0.1?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg8922] How has Solve changed in 3.0.1?
*From*: "John E. Derwent" <John.E.Derwent.1 at nd.edu>
*Date*: Sat, 4 Oct 1997 22:08:02 -0400
*Sender*: owner-wri-mathgroup at wolfram.com
It seems that in 2.2 Solve knew that exponentials are never 0, but it
doesn't in 3.0.1. Consider the problem of finding maxima and minima for
the function f[x,y]=(x^2+3y^2)e^(-x^2-y^2). The critical points are given
by
Solve[Grad[f[x,y]]==0],
i.e.,
Solve[-2x(-1+x^2+3y^2) e^(-x^2-y^2)==0, -2y(-3+x^2+3y^2) e^(-x^2-y^2)==0].
In 2.2 the output was the five solutions (0,0), (1,0), (-1,0), (0,1) and
(0,-1), after a message about using inverse functions.
In 3.0.1 the message about using inverse functions is there, there are also
some messages about not being able to verify some solutions, and that
limits may be necessary, and 16 more solutions are given. They are
(-Infinity, -I Infinity),
(-Infinity, I Infinity),
(Infinity, -I Infinity),
(Infinity,I Infinity),
(0, -Infinity),
(I Infinity, -Infinity) twice,
(0, Infinity),
(-I Infinity, Infinity) twice,
(-I Infinity, Infinity) twice,
(I Infinity, Infinity) twice,
(x, -Sqrt[3]Sqrt[1-y^2]),
(x, Sqrt[3]Sqrt[1-y^2]).
The last two are not solutions unless y=1 or -1 and x=0, and it is a strain
to consider the others as the solutions "in the limit". For example, why
not include (Infinity, Infinity)?
Does anyone know what has changed in Solve? I can find no explanation in
the book or in the Help. Everything there seems to suggest only finite
solutions. Of course the exponential term could be factored out before
solving, but why should that be necessary now when it wasn't before?
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